We discuss applications of the theory of quantum chaos to one of the paradigm models of many-body quantum physics — the Bose–Hubbard (BH) model, which describes, in particular, interacting ultracold Bose atoms in an optical lattice. After preliminary, pure quantum analysis of the system we introduce the classical counterpart of the BH model and the governing semiclassical equations of motion. We analyze these equations for the problem of Bloch oscillations (BOs) of cold atoms where a number of experimental results are available. The paper is written for nonexperts and can be viewed as an introduction to the field.
We discuss applications of the theory of quantum chaos to one of the paradigm models of many-body quantum physics — the Bose–Hubbard (BH) model, which describes, in particular, interacting ultracold Bose atoms in an optical lattice. After preliminary, pure quantum analysis of the system we introduce the classical counterpart of the BH model and the governing semiclassical equations of motion. We analyze these equations for the problem of Bloch oscillations (BOs) of cold atoms where a number of experimental results are available. The paper is written for nonexperts and can be viewed as an introduction to the field.
In Chapter 1 we set up the physical and mathematical framework for the description of interacting
bosons on a lattice. After a brief introduction to the basic concepts of Bose-Einstein condensation, and
how the condensate can be manipulated using optical lattices, we derive the Bose-Hubbard Hamiltonian
(BHH) which is the paradigm model as far as the quantum description is concerned. Then the semiclassical
limit of the BHH – the discrete nonlinear Schroedinger equation (DNLS) – as well as the mean-field
(classical) description, the so-called Gross-Pitaevskii equation (GPE), is discussed. We give an overview
of the relevant literature and of the various physical systems that are captured by the BHH including,
among others, arrays of Josephson junctions and bond vibrations in small molecules. The chapter ends
with an outline of the remaining thesis.
In Chapter 2, we look at some interesting experiments done in cold bosonic systems. Ostensibly,
done to realize the dc and ac Josephson effect in a system of cold bosons, it was soon found that the
system can show some rather interesting non-linear behavior. This includes features such as solitons
and breathers , but most interestingly, it shows the damping of the center-of-mass motion. Theoretical
explanations are given for both the experiments. The damping of center-of-mass of the system is seen as
a result of modulational instability. We give a different interpretation to this opening up lines for further
investigations. We outline these at the end of the chapter.
In Chapter 3, we discuss various experiments which contradicted the idea of a sharp modulational
instability. Rather, the loss of coherence is smeared out over a large domain. We then reproduce some
calculations from the literature showing how the dissipation can be thought of as arising from quantum
phase slips. We calculate the tunneling rates for quantum phase slips and show a phase diagram demarcating
stable and unstable regions of current which have been have obtained from mean-field calculations
via the Gutzwiller ansatz.
In Chapter 4, we look at the problem in details analyzing it as a Tomonaga-Luttinger liquid. The
framework of Tomonaga-Luttinger liquid is actually a non-perturbative tool through which we can investigate
an one-dimensional interacting bosonic fluid. We look at two possible mechanisms of decay
of current- We look at the effect of a weak link in a Josephson Junction array as well as local quantum
phase slips on a bosonic superfluid. We find that quantum phase slips are actually irrelevant under
renormalisation group in both cases and the supercurrent is revived.
In chapter 5, we look at the non-linear dynamics of the structures in greater detail, thus elucidating
chaos. This mechanism of chaos is different from all realizations of chaos known so far and is a result of
phase fluctuations, which are encoded by quantum phase slips.
The aim of this work is to study the dynamics of cold atoms trapped in a one-dimensional periodic lattice generated by an optical potential. Such a system is described by a standard model in condensed matter, namely the so-called Bose-Hubbard model. Arrays of cold neutral atoms or ions are the basic elements of some proposals of quantum computers, which motivates the interest of understanding their dynamics from a mathematical point of view. The main targets of this Master Thesis are the following. First, starting from a classical Hamiltonian of the Frenkel-Kontorova type and thanks to Perturbation Theory, to derive the discrete Bougolyubov-Gross-Pitaevskii (dBGP) equation. Second, starting from first principle formulation, to provide a systematic, quantum mechanical, perturbative derivation of the Bose-Hubbard (BH) model to orders higher than the first one in the small parameter.
Abstract This work develops a quantum control application of many-body quantum chaos for ultracold bosonic gases trapped in optical lattices. It is long known how to harness exponential sensitivity to changes in initial conditions for control purposes in classically chaotic systems. In the technique known as targeting, instead of a hindrance to control, the instability becomes a resource. Recently, this classical targeting has been generalized to quantum systems either by periodically countering the inevitable quantum state spreading or by introducing a control Hamiltonian, where both enable localized states to be guided along special chaotic trajectories toward any of a broad variety of desired target states. Only strictly unitary dynamics are involved; i.e. it gives a coherent quantum targeting. In this paper, the introduction of a control Hamiltonian is applied to Bose–Hubbard systems in chaotic dynamical regimes. Properly selected unstable mean field solutions can be followed particularly rapidly to states possessing precise phase relationships and occupancies. In essence, the method generates a quantum simulation technique that can access rather special states. The protocol reduces to a time-dependent control of the chemical potentials, opening up the possibility for application in optical lattice experiments. Explicit applications to custom state preparation and stabilization of quantum many-body scars are presented in one- and two-dimensional lattices (three-dimensional applications are similarly possible).
We study the Bloch dynamics of a quasi-one-dimensional Bose-Einstein condensate of cold atoms in a tilted optical lattice modeled by a Hamiltonian of Bose-Hubbard type. The corresponding mean-field system described by a discrete nonlinear Schr\"odinger equation can exhibit dynamical (or modulation) instability due to chaotic dynamics and equipartition over the quasimomentum modes. It is shown that these phenomena are related to Bogoliubov's depletion of the Bose-Einstein condensate and a decoherence of the condensate in the many-particle description. Three types of dynamics are distinguished: (i) decaying oscillations in the region of dynamical instability and (ii) persisting Bloch oscillations or (iii) periodic decay and revivals in the region of stability.
Experiments with ultracold atoms in optical lattice have become a versatile testing ground to study diverse quantum many-body Hamiltonians. A single-band Bose-Hubbard (BH) Hamiltonian was first proposed to describe these systems in 1998 and its associated quantum phase-transition was subsequently observed in 2002. Over the years, there has been a rapid progress in experimental realizations of more complex lattice geometries, leading to more exotic BH Hamiltonians with contributions from excited bands, and modified tunneling and interaction energies. There has also been interesting theoretical insights and experimental studies on “un- conventional” Bose-Einstein condensates in optical lattices and predictions of rich orbital physics in higher bands. In this thesis, I present our results on several multi- band BH models and emergent quantum phenomena. In particular, I study optical lattices with two local minima per unit cell and show that the low energy states of a multi-band BH Hamiltonian with only pairwise interactions is equivalent to an effec- tive single-band Hamiltonian with strong three-body interactions. I also propose a second method to create three-body interactions in ultracold gases of bosonic atoms in a optical lattice. In this case, this is achieved by a careful cancellation of two contributions in the pair-wise interaction between the atoms, one proportional to the zero-energy scattering length and a second proportional to the effective range. I subsequently study the physics of Bose-Einstein condensation in the second band of a double-well 2D lattice and show that the collision aided decay rate of the con- densate to the ground band is smaller than the tunneling rate between neighboring unit cells. Finally, I propose a numerical method using the discrete variable repre- sentation for constructing real-valued Wannier functions localized in a unit cell for optical lattices. The developed numerical method is general and can be applied to a wide array of optical lattice geometries in one, two or three dimensions.
The Bose-Hubbard model is the simplest model of interacting bosons on a lattice. It has recently been the focus of much attention due to the realization of this model with cold atoms in an optical lattice. The ability to tune parameters in the Hamiltonian as a function of time in cold atom systems has opened up the possibility of studying out-of-equilibrium dynamics, including crossing the quantum critical region of the model in a controlled way. In this paper, I give a brief introduction to the Bose Hubbard model, and its experimental realization and then give an account of theoretical and experimental efforts to understand out-of-equilibrium dynamics in this model, focusing on quantum quenches, both instantaneous and of finite duration. I discuss slow dynamics that have been observed theoretically and experimentally for some quenches from the superfluid phase to the Mott insulating phase and the picture of two timescales, one for fast local equilibration and another for slow global equilibration, that appears to characterize this situation. I also discuss the theoretical and experimental observation of the Lieb-Robinson bounds for a variety of quenches and the Kibble-Zurek mechanism in quenches from the Mott insulator to superfluid. I conclude with a discussion of open questions and future directions.